Structure Learning in Gaussian Graphical Models from Glauber Dynamics

TL;DR

This work addresses the problem of learning the edge structure of a Gaussian graphical model from dependent samples generated by Glauber dynamics. It introduces the first structure-learning algorithm for GGMs under this dynamic, proving that a time horizon of $T = \mathcal{O}\left(\frac{d^{2}\operatorname{polylog} p}{\beta_{\min}^{4}}\right)$ suffices to recover a graph with maximum degree $d$, and establishing a nearly matching minimax lower bound of $\Omega\left(d^{2}\log p\right)$ up to polynomial factors in $d$ and polylogarithmic factors in $p$. The algorithm relies on an edge-wise hypothesis test based on carefully designed update-interval statistics, including events $A_{ij}^{k}$ and $D_{ij}^{k}$ and a ratio statistic that isolates the edge influence $\beta_{ij}$ while controlling for unobserved neighbors and variance. The paper also proves information-theoretic lower bounds via Fano’s method, showing that the proposed method is nearly minimax optimal for a broad class of problems, and demonstrates that the computation can be parallelized to achieve $\tilde{\mathcal{O}}(d^{2}p)$ time. These results advance structure learning in non-i.i.d. settings with unbounded variables and have implications for learning dependence structures in social, biological, and financial networks where Glauber-type dynamics are natural.

Abstract

Gaussian graphical model selection is an important paradigm with numerous applications, including biological network modeling, financial network modeling, and social network analysis. Traditional approaches assume access to independent and identically distributed (i.i.d) samples, which is often impractical in real-world scenarios. In this paper, we address Gaussian graphical model selection under observations from a more realistic dependent stochastic process known as Glauber dynamics. Glauber dynamics, also called the Gibbs sampler, is a Markov chain that sequentially updates the variables of the underlying model based on the statistics of the remaining model. Such models, aside from frequently being employed to generate samples from complex multivariate distributions, naturally arise in various settings, such as opinion consensus in social networks and clearing/stock-price dynamics in financial networks. In contrast to the extensive body of existing work, we present the first algorithm for Gaussian graphical model selection when data are sampled according to the Glauber dynamics. We provide theoretical guarantees on the computational and statistical complexity of the proposed algorithm's structure learning performance. Additionally, we provide information-theoretic lower bounds on the statistical complexity and show that our algorithm is nearly minimax optimal for a broad class of problems.

Figures (1)

• Figure 1: Visualization of the idealized update sequence for vertices $i$ and $j$. At times $n_1$ and $n_3$, node $i$ is updated, while node $j$ is updated at time $n_2$. No updates occur for any other neighbors of $i$ in the interval between $n_0$ and $n_4$, except possibly $j$.

Theorems & Definitions (40)

• Lemma 1
• proof
• Definition 1: Event $A_{ij}^k$
• Lemma 2: Probability of event $A_{ij}^k$
• Definition 2: Event $D_{ij}^k$
• Lemma 3
• Lemma 4: Event C
• Definition 3: Event $B_j^k$
• Lemma 5: Conditional probability of event $B_{j}^k$
• Definition 4: Test statistic